Students can use a variety of representations to show their mathematical thinking. However, some representations might be more appropriate than others for a given task. We may not know which representations are most useful beforehand; often, some trial and error is required. In many cases, students will need to invent their own ways of representing their thinking before they are ready to use more conventional forms. This is an important step in helping students give meaning to the mathematics they are studying. It is especially important in the middle grades, as students begin to work with more abstract concepts .

However, it is our goal to move students from their unconventional representations to conventional forms that are important in communicating and extending mathematical ideas. In the first section, we looked at drawings and models as forms of representation. Now we will look at a different type of representation, drawing graphs, which will help students work with the concept of a variable.

In this section, you will use graphs to tell a mathematical story. The focus is on your own thought process, rather than your students'. As you complete the tasks, think about how this activity helps you gain a better understanding of the meaning of the graph and how the graph gives you a visual representation of the situations in the story.